Answer
No.
See counterexample below.
Work Step by Step
Build a counterexample using parallel vectors.
Let ${\bf u}$ =${\bf i, \ \ v}$ =${\bf -i,\ \ w}$ =${\bf 2i}.$
The cross products ${\bf u}\times{\bf v}$ and ${\bf u}\times{\bf w}$ both equal ${\bf 0}$
because the cross product of parallel vectors is the zero vector.
So, we have ${\bf u}\times{\bf v}={\bf u}\times{\bf w}, \quad {\bf u}\neq {\bf 0}$,
but ${\bf v}\neq {\bf w}$
